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Surface-tension-induced buckling of liquid-lined elastic tubes: a model pulmonary airway closure
No full textWe use a fully coupled, three-dimensional, finite-element method to study the evolution of the surface-tension-driven instabilities of a liquid layer that lines an elastic tube, a simple model for pulmonary airway closure. The equations of large-displacement shell theory are used to describe the deformations of the tube and are coupled to the Navier–Stokes equations, describing the motion of the liquid.
The liquid layer is susceptible to a capillary instability, whereby an initially uniform layer can develop a series of axisymmetric peaks and troughs, analogous to the classical instability that causes liquid jets to break up into droplets. For sufficiently high values of the liquid's surface tension, relative to the bending stiffness of the tube, the additional compressive load induced by the development of the axisymmetric instability can induce non-axisymmetric buckling of the tube wall. Once the tube has buckled, a strong destabilizing feedback between the fluid and solid mechanics leads to an extremely rapid further collapse and occlusion of the gas-conveying core of the tube by the liquid. We find that such occlusion is possible even when the volume of the liquid is too small to form an occluding liquid bridge in the axisymmetric tube
Hyperelliptic Addition Law
No full textGiven a family of genus g algebraic curves, with the equation f(x, y, ) = 0, we cosider two fiber-bundles U and X over the space of parameters . A fiber of U is the Jacobi variety of the curve. U is equipped with the natural groupoid structure that induces the canonical addition on a fiber. A fiber of X is the g-th symmetric power of the curve. We describe the algebraic groupoid structure on X using the Weierstrass gap theorem to define the àddition law' on its fiber. The addition theorems that are the subject of the present study are represented by the formulas, mostly explicit, dtermining the isomorphism of groupoids U X. At g=1 this gives the classic addition formulas for the elliptic Weierstrass and functions. To illustrate the efficiency of our approach the hyperelliptic curves of the form y2 = x2g+1 + 2g-1 i=0 4g+2-2ixi are considered. We construct the explicit form of the addition law for hyperelliptic Abelian vector functions and (the functions and form a basis in the field of hyperelliptic Abelian functions, i.e., any function from the field can be expressed as a rational function of and ). Addition formulas for the higher genera zetfunctions are discussed. The genus 2 result is written in a Hirota-like trilinear form for the sigma-function. We propose a conjecture to describe the general formula in these terms
On algebraic sets over metabelian groups
No full textWe investigate algebraic sets over certain finitely generated torsion-free metabelian groups. The class of groups under consideration is the class of so-called ρ-groups. It consists of all wreath products of finitely generated free abelian groups and their subgroups. In particular, it includes all free metabelian groups of finite rank. Our main result is a characterization of certain irreducible algebraic sets over ρ-groups. More precisely, we consider irreducible algebraic sets which are determined by a system of equations in n indeterminates. For their coordinate groups, we introduce a discrete invariant called the relative characteristic. This is an ordered pair of non-negative integers. We determine the structure of the coordinate group of the n-dimensional affine space, and show that its relative characteristic is (n, n). Then we characterize the irreducible algebraic sets of relative characteristic (n, n) and (0, k ) where 0 ≤ k ≤ n . We also obtain some examples of somewhat unusual algebraic sets over ρ-groups, thus demonstrating that algebraic sets over these groups are much more varied and complicated than, say, algebraic sets over free groups
Naïve noncommutative blowing up
No full textLet B(X,\mathscr{L},σ) be the twisted homogeneous coordinate ring of an irreducible variety X over an algebraically closed field k with dim X ≥ 2. Assume that c \in X and σ \in Aut(X) are in sufficiently general position. We show that if one follows the commutative prescription for blowing up X at c, but in this noncommutative setting, one obtains a noncommutative ring R = R(X,c,,σ) with surprising properties.
(1) R is always Noetherian but never strongly Noetherian
(2) If R is generated in degree one, then the images of the R-point modules in qgr-R are naturally in one-to-one correspondence with the closed points of X. However, in both qgr-R and gr-R, the R-point modules are not parametrized by a projective scheme.
(3) While qgr-R has finite cohomological dimension dim_k H^1 ( \mathscr{O} ) = ∞
Benchmark examples for model reduction of linear time invariant dynamical systems.
Get PDFWe present a benchmark collection containing some useful real world examples, which can be used to test and compare numerical methods for model reduction. All systems can be downloaded from the web and we describe here the relevant characteristics of the benchmark examples
Plancherel measure for GL(n,F) and GL(m,D): Explicit formulas and Bernstein decomposition
Get PDFLet F be a nonarchimedean local field, let D be a division algebra over F, let GL(n) = GL(n,F). Let denote Plancherel measure for GL(n). Let be a component in the Bernstein variety \Omega(\GL(n)). Then yields its fundamental invariants: the cardinality q of the residue field of F, the sizes m_1,..., m_t, exponents e_1,...,e_t, torsion numbers r_1,...,r_t\nu_{\Omega}\GL(n)$ and establish some new formal degree formulas. We derive, via the Jacquet-Langlands correspondence, the explicit Plancherel formula for GL(m,D)
Interpretation, coordination and conformity
No full textThe aim of this note is to investigate a very general problem of (radical) interpretation in terms of a simple coordination game: the conformity game. We show how, within our mathematical framework, the solution concept for the conformity game does indeed provide an algorithmic procedure facilitating triangulation, in the sense of Davidson
Parameter-free H(div) preconditioning for mixed finite element formulation of diffusion problems
No full textMixed finite element formulations of generalised diffusion problems yield linear systems with ill-conditioned, symmetric and indefinite coefficient matrices. Preconditioners with optimal work complexity that do not rely on artificial parameters are essential. We implement lowest order Raviart–Thomas elements and analyse practical issues associated with so-called ‘H(div) preconditioning’. Properties of the exact scheme are discussed in Powell & Silvester (2003, SIAM J. Matrix Anal. Appl., 25, 718–738). We extend the discussion, here, to practical implementation, the components of which are any available multilevel solver for a weighted H(div) operator and a pressure mass matrix. A new bound is established for the eigenvalue spectrum of the preconditioned system matrix and extensive numerical results are presented
A hyperbolic well-posed model for the flow of granular materials
No full textA plasticity model for the flow of granular materials is presented which is derived from a physically based kinematic rule and which is closely related to the double-shearing model, the double-sliding free-rotating model and also to the plastic-potential model. All of these models incorporate various notions of the concept of rotation-rate and the crucial idea behind the model presented here is that it identifies this rotation-rate with a property associated with a Cosserat continuum, namely, the intrinsic spin. As a consequence of this identification, the stress tensor may become asymmetric. For simplicity, in the analysis presented here, the material parameters are assumed to be constant. The central results of the paper are that (a) the model is hyperbolic for two-dimensional steady-state flows in the inertial regime and (b) the model possesses a domain of linear well-posedness. Specifically, it is proved that incompressible flows are well-posed